Has anyone thought about making a neural network ask questions, instead of only answering them?
Maybe neural networks are not the best tool for this.

It seems to me that an equivalent of the your notion of 'a question to help the classification' would be to use Machine Learning (ML) to obtain a human-readable ruleset which performs the classification. The idea is that, if you follow an applicable chain of rules all the way through to the end, you have a classifier, if you stop before that, you have an indicator of which features of the input give more coarse-grained classifications, which can be seen as a progressively detailed sequence of questions that 'help the classification'.



More detail on various options for using ML to create rulesets can be found in my answer to this question.